Відносне зростання рядів Діріхле з різними абсцисами абсолютної збіжності
Анотація
УДК 517.537.72
We study the growth of a Dirichlet series $F(s)=\sum _{n=1}^{\infty}f_n\exp\{s\lambda_n\}$ with zero abscissa of absolute convergence with respect to the entire Dirichlet series $G(s)=\sum _{n=1}^{\infty}g_n\exp\{s\lambda_n\}$ by using the generalized quantities of order $\varrho^0_{\beta,\beta}[F]_G=\varlimsup\nolimits_{\sigma\uparrow 0}\dfrac{\beta(M^{-1}_G(M_F(\sigma)))}{\beta(1/|\sigma|)}$ and lower order $\lambda^0_{\beta,\beta}[F]_G=\varliminf_{\sigma\uparrow 0} \dfrac{\beta(M^{-1}_G(M_F(\sigma)))}{\beta(1/|\sigma|)},$ where $M_F(\sigma)=\sup\{|F(\sigma+it)|\colon t\in{\Bbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(\sigma),$ and $\beta$ is a positive increasing function growing to $+\infty.$
Посилання
Ch. Roy, On the relative order and lower order of an entire functiion, Bull. Calcutta Math. Soc., 102, № 1, 17 – 26 (2010).
S. K. Data, A. R. Maji, Relative order of entire functions in terms of their maximum terms, Int. J. Math. and Anal., 5, № 43, 2119 – 2126 (2011).
S. K. Data, T. Biswas, Ch. Ghosh, Growth analysis of entire functions concerning generalized relative type and generalized relative weak type, Facta Univ. Ser. Math. and Inform., 30, № 3, 295 – 324 (2015).
S. K. Data, T. Biswas, A. Hoque, Some results on the growth analysis of entire function using their maximum terms and relative $L^{ast}$ -order, J. Math. Ext., 10, № 2, 59 – 73 (2016).
S. K. Data, T. Biswas, P. Das, Some results on generalized relative order of meromorohic functions, Ufa Math. J., 8,№ 2, 92 – 103 (2016), https://doi.org/10.13108/2016-8-2-95 DOI: https://doi.org/10.13108/2016-8-2-95
S. K. Data, T. Biswas, Growth analysis of entire functions of two complex variables, Sahad Commun. Math. Anal.,3, Issue 2, 13 – 22 (2016).
S. K. Data, T. Biswas, Some growth analysis of entire functions in the form of vector valued Dirichlet series on the basis on their relative Ritt $L^{ast}$-order and relative Ritt $L^{ast}$-lower order, New Trends Math. Sci., 5, № 2, 97 –103 (2017).
Ya. D. P`yany`lo, M. N. Sheremeta, O roste czely`kh funkczij, predstavlenny`kh ryadami Dirikhle, Izv. vuzov. Matematika, № 10, 91 – 93 (1975).
O. M. Mulyava, M. M. Sheremeta, Relative growth of Dirichlet series, Mat. Stud., 49, № 2, 158 – 164 (2018), https://doi.org/10.15330/ms.49.2.158-164 DOI: https://doi.org/10.15330/ms.49.2.158-164
O. M. Mulyava, M. M. Sheremeta, Remarks to relative growth of entire Dirichlet series, Visnyk Lviv Univ. Ser. Mech., Math., № 87, 73 – 81 (2019).
Yu. M. Gal`, M. N. Sheremeta, O roste analiticheskikh v poluploskosti funkczij, zadanny`kh ryadami Dirikhle, Dokl. AN USSR, ser., № 12, 1065 – 1067 (1978).
J. F. Ritt, On certain points in the theory of Dirichlet series, Math. Amer. J., 50, 73 – 83 (1928), https://doi.org/10.2307/2370849 DOI: https://doi.org/10.2307/2370849
A. M. Gajsin, Oczenki rosta funkczij, predstavlenny`kh ryadami Dirikhle v poluploskosti, Mat. sb, 117, № 3, 412 – 424 (1982).
M. M. Sheremeta, Asimptoticheskoe povedenie czely`kh funkczij, zadanny`kh stepenny`mi ryadami i ryadami Dirikhle, Diss. . . . d-ra fiz.-mat. nauk. – Kiev (1987).
Yu. M. Gal`, O roste analiticheskikh funkczij, zadanny`kh absolyutno skhodyashhimisya v poluploskosti ryadami Dirikhle, Drogoby`ch (1980). – 30 с. – Dep. v VINITI, 1980, № 4080-80 Dep.
A. G. Azpeitia, A remark on the Ritt order of entire functions defined by Dirichlet series, Proc. Amer. Math. Soc., 12, 722 – 723 (1961), https://doi.org/10.2307/2034864 DOI: https://doi.org/10.2307/2034864
A. G. Azpeitia, On the lower linear type of entire functions defined by Dirichlet series, Bull. Unione Mat. Ital. A, 15, № 3, 635 – 638 (1978).
V. S. Bojchuk, O roste ryadov Dirikhle, absolyutno skhodyashhikhsya v poluploskosti, Mat. sb., Nauk. dumka, Kiev (1976), p. 238 – 240.
M. M. Sheremeta, S. I. Fedy`nyak, O proizvodnoj ryada Dirikhle, Sib. mat. zhurn., 39, № 1, 206 – 223 (1998).
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